Quantum Error Correction Shyam Sundhar R Department of EE Mid Term - - PowerPoint PPT Presentation

Quantum Error Correction

Shyam Sundhar R

Department of EE

Mid Term Presentation, CS 682

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 1 / 32

Outline

1

Shor Code Decoherence Tackling Decoherence

2

Error Correction by mapping into 2D spaces Density Operator The theory

3

Interaction Interaction Operator Recovery

4

Fundamentals of error correcting codes Basic Definitions Recovery Operator State Independent Error Modeling

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 2 / 32

Outline

1

Shor Code Decoherence Tackling Decoherence

2

Error Correction by mapping into 2D spaces Density Operator The theory

3

Interaction Interaction Operator Recovery

4

Fundamentals of error correcting codes Basic Definitions Recovery Operator State Independent Error Modeling

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 3 / 32

What is decoherence?

How do we model it?

Change in phase relation of different states due to environmental interaction.

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 4 / 32

What is decoherence?

How do we model it?

Change in phase relation of different states due to environmental interaction. |e0|0 = (|a0|0 + |a1|1) |e0|1 = (|a2|0 + |a3|1) where |e0 is the initial state of the environment and |a0,|a1,|a2,|a3 are the final states of the environment.

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 4 / 32

Outline

1

Shor Code Decoherence Tackling Decoherence

2

Error Correction by mapping into 2D spaces Density Operator The theory

3

Interaction Interaction Operator Recovery

4

Fundamentals of error correcting codes Basic Definitions Recovery Operator State Independent Error Modeling

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 5 / 32

encoding of qubits

|0S =

1 2 √ 2(|000 + |111) ⊗ (|000 + |111) ⊗ (|000 + |111

|1S =

1 2 √ 2(|000 − |111 ⊗ (|000 − |111) ⊗ (|000 − |111

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 6 / 32

encoding of qubits

|0S =

1 2 √ 2(|000 + |111) ⊗ (|000 + |111) ⊗ (|000 + |111

|1S =

1 2 √ 2(|000 − |111 ⊗ (|000 − |111) ⊗ (|000 − |111

Decoherence of the first qubit (|0S− > 1/ √ 2)[(|a0|0 + |a1|1)|00 + (|a2|0 + |a3|1)|11)] (|1S− > 1/ √ 2)[(|a0|0 + |a1|1)|00 − (|a2|0 + |a3|1)|11)]

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 6 / 32

encoding of qubits

1/2 √ 2(|a0 + |a3)(|000 + |111)+ 1/2 √ 2(|a0 − |a3)(|000 − |111)+ 1/2 √ 2(|a1 + |a2)(|100 + |011)+ 1/2 √ 2(|a1 − |a2)(|100 − |011)

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 7 / 32

encoding of qubits

1/2 √ 2(|a0 + |a3)(|000 + |111)+ 1/2 √ 2(|a0 − |a3)(|000 − |111)+ 1/2 √ 2(|a1 + |a2)(|100 + |011)+ 1/2 √ 2(|a1 − |a2)(|100 − |011) 1/2 √ 2(|a0 + |a3)(|000 − |111)+ 1/2 √ 2(|a0 − |a3)(|000 + |111)+ 1/2 √ 2(|a1 + |a2)(|100 − |011)+ 1/2 √ 2(|a1 − |a2)(|100 + |011)

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 7 / 32

Projection of states in bell basis states(|abc + |a

′b ′c ′).

Use of ancilla qubits to differentiate each subspace. Measurement of ancilla qubit. Application of appropriate operator.

Takeaway

The important thing to note is that the state of the environment is the same for corresponding vectors from the decoherence of the two quantum states encoding 0 and encoding 1. Since the corresponding vectors are same If it is projected to the superposition of the corresponding bell basis states appropriately the final state after correction tend to be in superposition For instance one projection would be to 1/2 √ 2(|a0 − |a3)(|000 − |111) + 1/2 √ 2(|a0 − |a3)(|000 + |111)

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 8 / 32

Outline

1

Shor Code Decoherence Tackling Decoherence

2

Error Correction by mapping into 2D spaces Density Operator The theory

3

Interaction Interaction Operator Recovery

4

Fundamentals of error correcting codes Basic Definitions Recovery Operator State Independent Error Modeling

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 9 / 32

What is density operator?

Density operators

Positive semidefinite operators having trace equal to 1 are called density

• perators.

D(X) = ρ ∈ Pos(X) : Tr(ρ) = 1

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 10 / 32

What is density operator?

Density operators

Positive semidefinite operators having trace equal to 1 are called density

• perators.

D(X) = ρ ∈ Pos(X) : Tr(ρ) = 1 D(X) will be used to denote the collection of density operators acting

• n a complex Euclidean space X .

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 10 / 32

What is density operator?

Density operators

Positive semidefinite operators having trace equal to 1 are called density

• perators.

D(X) = ρ ∈ Pos(X) : Tr(ρ) = 1 D(X) will be used to denote the collection of density operators acting

• n a complex Euclidean space X .

For any complex Euclidean space X , the set D(X ) of density

• perators acting on X is convex and compact. The extreme points of

D(X ) coincide with the rank-one projection operators. These are the

• perators of the form uu∗ for u ∈ S(X ) being a unit vector.

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 10 / 32

Quantum States as Density operators

Definition

A quantum state is a density operator of the form ρ ∈D(X ) for some choice of a complex Euclidean space X

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 11 / 32

Quantum States as Density operators

Definition

A quantum state is a density operator of the form ρ ∈D(X ) for some choice of a complex Euclidean space X

Pure state

A quantum state ρ ∈ D(X) is said to be a pure state if it has rank equal to 1. Equivalently, ρ is a pure state if there exists a unit vector u ∈ X such that ρ = uu∗

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 11 / 32

Quantum States as Density operators

Definition

A quantum state is a density operator of the form ρ ∈D(X ) for some choice of a complex Euclidean space X

Pure state

A quantum state ρ ∈ D(X) is said to be a pure state if it has rank equal to 1. Equivalently, ρ is a pure state if there exists a unit vector u ∈ X such that ρ = uu∗

Theorem

state representation ρ = Σp(a)ρa

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 11 / 32

Quantum States as Density operators

Definition

A quantum state is a density operator of the form ρ ∈D(X ) for some choice of a complex Euclidean space X

Pure state

A quantum state ρ ∈ D(X) is said to be a pure state if it has rank equal to 1. Equivalently, ρ is a pure state if there exists a unit vector u ∈ X such that ρ = uu∗

Theorem

state representation ρ = Σp(a)ρa

Unitary Operation

Any state ρf after Unitary Operation Aa can be written as ρf = AaρiA∗

a

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 11 / 32

Outline

1

Shor Code Decoherence Tackling Decoherence

2

Error Correction by mapping into 2D spaces Density Operator The theory

3

Interaction Interaction Operator Recovery

4

Fundamentals of error correcting codes Basic Definitions Recovery Operator State Independent Error Modeling

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 12 / 32

the main idea

mapping |ψ into a higher dimensional Hilbert space (using ancilla qubits which are assumed to be in their |0 states initially): (α|0 + β|1|000...α|0L + β|1L. error induced maps the new state into one of a family of two-dimensional subspaces which preserve the relative coherence of the quantum information (i.e. in each subspace, the state of the computer should be in a tensor product state with the environment). A measurement is then performed which projects the state into one of these subspaces

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 13 / 32

Outline

1

Shor Code Decoherence Tackling Decoherence

2

Error Correction by mapping into 2D spaces Density Operator The theory

3

Interaction Interaction Operator Recovery

4

Fundamentals of error correcting codes Basic Definitions Recovery Operator State Independent Error Modeling

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 14 / 32

Modeling of interaction

Any state ρi after interaction can be written as ρf = ΣaAaρiA∗

a

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 15 / 32

Modeling of interaction

Any state ρi after interaction can be written as ρf = ΣaAaρiA∗

a

For orthonormal basis µ of environment, evolution operator U and initial environmental state |e, Aa = µa|U|e and ΣaA∗

aAa = I

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 15 / 32

Modeling of interaction

Any state ρi after interaction can be written as ρf = ΣaAaρiA∗

a

For orthonormal basis µ of environment, evolution operator U and initial environmental state |e, Aa = µa|U|e and ΣaA∗

aAa = I

Aa are linear operators of the hilbert space of the system acting due to environment and they are called interaction operators

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 15 / 32

Outline

1

Shor Code Decoherence Tackling Decoherence

2

Error Correction by mapping into 2D spaces Density Operator The theory

3

Interaction Interaction Operator Recovery

4

Fundamentals of error correcting codes Basic Definitions Recovery Operator State Independent Error Modeling

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 16 / 32

Conditions for recovery

Necessary and sufficient condition

0L|A∗

aAb|1L = 0

, 0L|A∗

aAb|0L = 1L|A∗ aAb|1L

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 17 / 32

Conditions for recovery

Necessary and sufficient condition

0L|A∗

aAb|1L = 0

, 0L|A∗

aAb|0L = 1L|A∗ aAb|1L

logical zero and logical one go to orthogonal states

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 17 / 32

Conditions for recovery

Necessary and sufficient condition

0L|A∗

aAb|1L = 0

, 0L|A∗

aAb|0L = 1L|A∗ aAb|1L

logical zero and logical one go to orthogonal states length and inner products of the projections of the corrupted logical zero and one should be the same

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 17 / 32

Conditions for recovery

Necessary and sufficient condition

0L|A∗

aAb|1L = 0

, 0L|A∗

aAb|0L = 1L|A∗ aAb|1L

logical zero and logical one go to orthogonal states length and inner products of the projections of the corrupted logical zero and one should be the same

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 17 / 32

Example

decoherence takes the form |ψi = α|0 + β|1− > ρf = αα∗ αβ∗e−γ αβ∗e−γ ββ∗

• Shyam Sundhar R

(IIT K) Quantum Error Correction Mid Term Presentation, CS 682 18 / 32

Example

decoherence takes the form |ψi = α|0 + β|1− > ρf = αα∗ αβ∗e−γ αβ∗e−γ ββ∗

• if decoherence occurs in the following fashion

|e|0− > |e0|0 |e|1− > |e1|1 where e0|e1 = e−γ. Using |µ0 = |e0 and (|µ1 = |e1 − e−γ|e0)/ √ 1 − e−2γ we obtain A0 = 1 e−γ

• A1 =

√ 1 − e−2γ

• Shyam Sundhar R

(IIT K) Quantum Error Correction Mid Term Presentation, CS 682 18 / 32

In a different basis

To understand correction of this type of error we choose a different basis of the environment

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 19 / 32

In a different basis

To understand correction of this type of error we choose a different basis of the environment |µ+ = (|e0 + |e1)/

• 2(1 + e−γ)

|µ− = (|e0 − |e1)/

• 2(1 − e−γ)

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 19 / 32

In a different basis

To understand correction of this type of error we choose a different basis of the environment |µ+ = (|e0 + |e1)/

• 2(1 + e−γ)

|µ− = (|e0 − |e1)/

• 2(1 − e−γ)

A+ = a+ 1 1

• ; A− = a−

1 −1

• where a+ =
• (1 + e−γ)/2 and a− =
• (1 − e−γ)/2

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 19 / 32

Correction

Encoding of a state

|0L = (|0 + |1)(|0 + |1)(|0 + |1) |1L = (|0 − |1)(|0 − |1)(|0 − |1)

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 20 / 32

Correction

Encoding of a state

|0L = (|0 + |1)(|0 + |1)(|0 + |1) |1L = (|0 − |1)(|0 − |1)(|0 − |1) A+|0L = (|0 + |1)(|0 + |1)(|0 + |1) A1

−|0L = (|0 − |1)(|0 + |1)(|0 + |1)

A2

−|0L = (|0 + |1)(|0 − |1)(|0 + |1)

A3

−|0L = (|0 + |1)(|0 + |1)(|0 − |1)

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 20 / 32

Correction

Encoding of a state

|0L = (|0 + |1)(|0 + |1)(|0 + |1) |1L = (|0 − |1)(|0 − |1)(|0 − |1) A+|0L = (|0 + |1)(|0 + |1)(|0 + |1) A1

−|0L = (|0 − |1)(|0 + |1)(|0 + |1)

A2

−|0L = (|0 + |1)(|0 − |1)(|0 + |1)

A3

−|0L = (|0 + |1)(|0 + |1)(|0 − |1)

A+|1L = (|0 − |1)(|0 − |1)(|0 − |1) A1

−|1L = (|0 + |1)(|0 − |1)(|0 − |1)

A2

−|1L = (|0 − |1)(|0 + |1)(|0 − |1)

A3

−|1L = (|0 − |1)(|0 − |1)(|0 + |1)

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 20 / 32

Recovery operators will be R+ = (|0L0L| + |1L1L|) R1

− = (|0L0L| + |1L1L|)σ1 z

R2

− = (|0L0L| + |1L1L|)σ2 z

R3

− = (|0L0L| + |1L1L|)σ3 z

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 21 / 32

Outline

1

Shor Code Decoherence Tackling Decoherence

2

Error Correction by mapping into 2D spaces Density Operator The theory

3

Interaction Interaction Operator Recovery

4

Fundamentals of error correcting codes Basic Definitions Recovery Operator State Independent Error Modeling

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 22 / 32

We want to preserve a k-dimensional subspace against some known

• errors. This is accomplished by mapping the states into a larger,

n-dimensional Hilbert space. We define an (n,k)-quantum code as a k-dimensional subspace of an n-dimensional Hilbert space. The symbol C is used for the code An encoding operator for C is a unitary operator E from a k-dimensional Hilbert space Q onto C. A decoding operator is a right inverse of an encoding operator.

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 23 / 32

Usually k = 2d and n = 2r The encoding operator can be implemented as a unitary operator

• nQ⊗d ⊗ Q⊗rd ⊗ Q⊗a

the last factor has a ancillary qubits whose state before and after the

• peration is intended to be |0.

The ancillas can be used as scratch pad memory during the process of measurement needed to recover C

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 24 / 32

A quantum error-correcting code is a pair (C, R) consisting of a quantum code and a recovery operator. The correcting properties of an error correcting code depend on the interaction with the environment. Let A be a family of linear operators as described previously. The fidelity of the code is determined by the fidelity of the composition RA restricted to C. The fidelity of the error-correcting code is thus defined as F(C, RA) = min|ψ∈CF(|ψ, RA) = min|ψ∈CΣr,a|ψ|RrAa|ψ|2

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 25 / 32

The error of the code is defined as E(C, RA) = max|ψ∈CΣr,a|(RrAa − ψ|RrAa|ψ)|ψ|2

Theorem

The operator Aa is in A(C, R) if when restricted to C, RrAa = λraI for each Rr ∈ R.

Theorem

The code C can be extended to an A-correcting code if for all basis elements|iL, |jL(i = j) and operators Aa, Ab in A iL|A∗

aAb|iL = jL|A∗ aAb|jL

, iL|A∗

aAb|jL = 0

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 26 / 32

Outline

1

Shor Code Decoherence Tackling Decoherence

2

Error Correction by mapping into 2D spaces Density Operator The theory

3

Interaction Interaction Operator Recovery

4

Fundamentals of error correcting codes Basic Definitions Recovery Operator State Independent Error Modeling

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 27 / 32

Constructing a Recovery operator

Let Υi the subspace spanned by Aa|iL (for all a). The Υi are

• rthogonal subspaces. Let |vi

r be an orthonormal basis for Υi. Since

|vi

r are mutually orthogonal there exist unitary Vr which return |vi r

to the corresponding state |iL Vr|vi

r = |iL

The recovery operator is given by the interaction operators R = O, R1, ..., Rr, ... where O is the projection onto the orthogonal complement of L i V i , i.e. the part of the Hilbert space which is not reached by acting on the code with the Aa, and Rr = VrΣi|vi

rvi r|

That R is a superoperator follows from the observation that it is a sum of orthogonal projections followed by unitary operators where the projections span the Hilbert space.

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 28 / 32

To ensure that R recovers the state, we need unitary operators Ui such that Ui|v0

r = |vi r and for all Aa, UiAa|0L = Aa|iL.

The existence of unitary operators satisfying the second condition follows from the theorem according to which the innerproduct relationships between the Aa|0L and the Aa|iL are identical. Given such Ui, |vi

r can be made to satisfy the remaining condition by

choosing the basis |v0

r and defining |vi r = Ui|v0 r

Given the above conditions it can be shown that RrAa|ψ = β0

ar|ψ

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 29 / 32

Outline

1

Shor Code Decoherence Tackling Decoherence

2

Error Correction by mapping into 2D spaces Density Operator The theory

3

Interaction Interaction Operator Recovery

4

Fundamentals of error correcting codes Basic Definitions Recovery Operator State Independent Error Modeling

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 30 / 32

Isomorphism

Theorem

C is an A-correcting code if there is an isomorphism ρ : H ≃ C ⊗ E ⊕ D such that for all Aa ∈ A and |ψ ∈ C, Aa|ψ = ρ(|ψ ⊗ |E(a) for some vector |E(a) depending on Aa alone. The idea is to ensure that the effect of the environment is clearly separated from the state to be preserved. Thus E takes up all the information from the environment and the final state in E encodes the environments effect on the code. The final state in E is called the error syndrome. D is the summand of H which is normally never reached by A.

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 31 / 32

Realisation

Let R be described by the interaction operators (V0P0, ..., VrmPrm), where the Pr are projections onto the orthogonal subspaces Υr, and the Vr are unitary.

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 32 / 32

Realisation

Let R be described by the interaction operators (V0P0, ..., VrmPrm), where the Pr are projections onto the orthogonal subspaces Υr, and the Vr are unitary. Let M be a separate (ancillary) system with standard basis |rM .

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 32 / 32

Realisation

Let R be described by the interaction operators (V0P0, ..., VrmPrm), where the Pr are projections onto the orthogonal subspaces Υr, and the Vr are unitary. Let M be a separate (ancillary) system with standard basis |rM . Let Wr be a unitary operator on M with the property that Wr|0M = |rM (i.e. Wr is a unitary extension of |rM0M|).

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 32 / 32

Realisation

Let R be described by the interaction operators (V0P0, ..., VrmPrm), where the Pr are projections onto the orthogonal subspaces Υr, and the Vr are unitary. Let M be a separate (ancillary) system with standard basis |rM . Let Wr be a unitary operator on M with the property that Wr|0M = |rM (i.e. Wr is a unitary extension of |rM0M|). The operator S = ΣrPr ⊗ Wris unitary and has the property that Υr ⊗ |0M goes to Υr ⊗ |rM. (This is a generalization of the standard controlled-not operations in quantum computing.)

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 32 / 32

Realisation

Let R be described by the interaction operators (V0P0, ..., VrmPrm), where the Pr are projections onto the orthogonal subspaces Υr, and the Vr are unitary. Let M be a separate (ancillary) system with standard basis |rM . Let Wr be a unitary operator on M with the property that Wr|0M = |rM (i.e. Wr is a unitary extension of |rM0M|). The operator S = ΣrPr ⊗ Wris unitary and has the property that Υr ⊗ |0M goes to Υr ⊗ |rM. (This is a generalization of the standard controlled-not operations in quantum computing.) If M starts in the state |0M , then we can perform R by first applying S , then measuring M in the standard basis and finally applying Ur to the coding space if the outcome of the measurement is|rM Here Pr can be thought of as Σi|vi

rvi r|

Shyam Sundhar R (IIT K) Quantum Error Correction Mid Term Presentation, CS 682 32 / 32